Seismic data are typically gathered using an array of detectors. In the case of marine data, hydrophones measure pressure fluctuations in the water caused by incoming seismic waves. Geophones measure vector quantities such as displacement, velocity or acceleration. In the case of marine data, a plurality of cables or streamers, which are spaced apart typically by about 100 metres, are towed behind a boat. Each cable has detectors spaced along the cable at intervals. In the case of land data, a geophone array is laid out on the ground with the geophones in an approximate grid formation. The detector array detects seismic signals from reverberations of a signal from a seismic source, such as an airgun for marine data. However, the detectors will not be exactly regularly spaced in the array. In particular, in the case of marine data, the cables drift and so spacing between the cables becomes irregular and changes the distances between the cables and the offsets of the detectors from the source. In Ocean Bottom (OBC or OBS) acquisition, a detector array is fixed on the sea bed. The source may be an airgun mounted on a boat.
Often, in the processing of seismic data it is necessary or desirable to spatially interpolate data. This may be because there are obstacles to the acquisition such as man made or topological obstacles in land data or structures such as an oil rig in marine data, or there may be gaps in the acquisition due to instrument failure. It also may be desirable to interpolate the data simply to decrease the sampling intervals for future processing steps and to increase the resolution. There are cost considerations and other practical considerations which limit the number of receivers which can be deployed in acquisition. In particular, in the acquisition of marine data, there is a practical limit to how closely spaced the cables or streamers may be without risk of entanglement. It therefore may be desirable to obtain extra lines of data intermediate between the streamers by interpolation.
It also may be desirable to simplify future processing steps to “regularise” data ie to take irregularly spaced data and interpolate to shift the data points to be regularly spaced.
Yen, J. L., (On Nonuniform Sampling of Bandwidth-Limited Signals, IRE Transactions on Circuit Theory 1956, 3, 251-257) discloses methods for interpolating non-uniformly sampled signals and in particular Yen's fourth theorem can be applied to the interpolation of seismic data. However, this method has its limitations due to aliasing. Chen, David Shi and Allebach, Jan P., 1987, “Analysis of Error in Reconstruction of Two-Dimensional Signals from Irregularly Spaced Samples”, IEEE Transactions on Acoustics, Speech and Signal Processing, 35, 173-180 discloses an extension to Yen's fourth theorem.
Recent developments, in marine acquisition record not only the pressure using hydrophones but also record particle velocity or acceleration in directions parallel to the surface.